Deterministic and Stochastic Prey–Predator Model for Three Predators and a Single Prey
Round 1
Reviewer 1 Report
The paper is interesting and the results are original. I recommend the publication of the paper.
Author Response
Dear Professors,
I am deeply grateful to you for your careful reading and useful comments and suggestions that have led to a significant improvement of the earlier version of my manuscript.
Thank you again
Sincerely yours
Reviewer 2 Report
The authors considered a population model with three predators and one prey, where they studied the influence of harvesting and stochastic fluctuations. In the first part of the paper a stability analysis of the deterministic system without noise is performed, while in the second part the influence of noise and, separately, the influence of stochastic fluctuations of the harvesting parameters is analyzed.
While the analytical and numerical results appear to be correct, there are several issues of the presentation and the interpretation of the results as well as of the conclusions drawn from them, which must be addressed before I can recommend this work for publication:
The existing literature on this topic is not properly presented and discussed. The whole literature overview in the introduction consists of only three sentences:
“Lotka and Volterra independently created two types of prey-predatory model known as the "Lotka-Volterra model" [1, 2]. Since then, many scientists have modified and developed the Lotca-Volterra model to accurately describe the ecosystem. Numerous studies have examined the case of the presence of more than one predator [2, 3, 4, 5, 6, 7, 8, 9, 10, 11].”
Due to the lack of a proper discussion of the existing literature, it is very difficult for a potential reader to guess what one can lean from this paper, what is already known and what are the new results.
The analysis partly is very similar to the analysis performed in [24]. Without a proper discussion of common features and differences between the system considered in the submitted paper and the system considered in [24] it is difficult to value the gain of knowledge. To me it seams that the authors added an additional predator (y(t)) to the system considered in [24] and modified the interference of the predators. They performed a similar stability analysis and a similar analysis of the effect of stochastic fluctuations of the harvesting parameters. Additionally, a bifurcation analysis of the deterministic system and the influence of noise was considered.
The relation between the systems should be discussed properly. Additionally, the following questions should be answered:
Which influence has the additional predator on the systems dynamics compared to the system in [24]?
Which influence has the changed interference of the predators compared to the system in [24]?
For certain parameters, the system in the submitted paper equals the system in [24]. What happens to the equilibria and the presented bifurcation scenarios in this limit?
The terms of the considered model, Eq. (1), should be properly motivated. See, for example, the introduction of Section 2 in [24].
No conclusions are drawn from the results of Section 5.1. What can the potential reader learn from the results derived there?
The authors proof that there is a transcritical bifurcation with respect to parameter q_1. In contrast, in Section 6 a supercritical Hopf bifurcation with respect to parameter q_3 is mentioned, which is visualized in Fig. 2 without further explanation. How can the authors be sure that this is a Hopf bifurcation? A proof, an explanation, or a reference should be added to justify this statement.
Are there bifurcations with respect to q_2?
On page 14 the authors state: “In the stochastic model (12), the conditions of Theorem. ?? are verified; thus, the number of the three predators will go to zero exponentially almost surely as indicated in Fig. 6. If one gradually increases the intensities of sigma_i = 0.9 and keeping the remaining parameters unchanged, the system (6) loses its stability as shown in Fig. 6.”. This statement is not justified properly or even wrong, since, in the bottom panel of Fig. 6, y(t), z(t), w(t) tend to zero just as in the top panel.
The paper contains many typos, missing labels and grammatical errors, which should be corrected.
Author Response
Dear Professors,
We are deeply grateful to the reviewers for their careful reading and valuable comments and suggestions that have significantly improved the earlier version of our manuscript.
We have considered all comments raised in their report and all suggested corrections.
Please see the PDF file?
Thank you again
Sincerely yours
Author Response File: 
Author Response.pdf
Round 2
Reviewer 2 Report
The authors have appropriately addressed all of my comments.
However, before the paper is ready for publication the following minor issues should be resolved:
The strong fluctuations in Fig. 4 below the bifurcation point (q_2<0.4) should be discussed. Are they caused by a low precision of the numerical method?
The black curves in Fig. 7 should be explained and added to the legend of the figure.
Axis labels should be added to Figures 7 and 8 (as in the previous version of the manuscript).
Author Response
"Please see the attachment."
Author Response File: Author Response.pdf
